Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×C22⋊((C112⋊C3)⋊C5) |
| unambiguous semidirect product | (C22×C3)⋊((C112⋊C3)⋊C5) |
| unambiguous semidirect product | (C3×#10597243)⋊A4 |
| unambiguous semidirect product | (C3×C22⋊(C112⋊C3))⋊C5 |
| Fitting, socle | C22×C3×C112 ⋊ C3×C5 |
| derived, Hall | C22×C112 ⋊ C32×C5 |
| Hall | C3×C22⋊(C112⋊C3) ⋊ C5 |
| Hall | C22×#10597243 ⋊ C32 |
| Hall | #10597243 ⋊ C3×A4 |
| Sylow | C112 ⋊ C3×C5×A4 |
| Sylow | C22 ⋊ C3×(C112⋊C3)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C22⋊((C112⋊C3)⋊C5) |
| commutator, cosocle, hypocenter | C22×C112 | C32×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C112 | C3×C5 |
| LCS | G ▷ C22×C112 | |
| UCS | C1 ◁ C3 | |
136 conjugacy classes of subgroups.
156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 521);